In addition to a viewpoint on identity types and a general class of categorical models, homotopy type theory is characterized by new homotopically motivated axioms and type-theoretic structures. Notable among these are:

Properties

Advantages

As a foundation for mathematics, homotopy type theory (also called univalent foundations) has the following advantages. Many of these advantages are shared with some other foundational systems, but no other known system shares all of these, and some are unique to HoTT.

It inherits the good computational properties of intensional Martin-Löf type theory. Some of its new axioms, such as univalence and function extensionality, are not fully understood yet from a computational perspective, but progress is being made. [This paragraph should be updated in view of the computational interpretation of univalence by the cubical model.]

It is constructive by default, but can easily be made classical by adding axioms. This makes it potentially more expressive at essentially no cost. (In fact, it is not entirely clear how possible it is to do homotopy theory constructively in other foundations.)

It can (conjecturally) be internalized in many categories and higher categories, providing an internal logic which enables a single proof to be reinterpreted in many places with many different meanings.

It is naturally isomorphism- and equivalence-invariant (non-evil). This is a consequence of the univalence axiom: any property or structure (even one which speaks only about sets and makes no reference to homotopy theory) which is expressible in the theory must be invariant under isomorphism/equivalence.

Notions such as propositions and sets are defined objects, which inherit good computational properties from the underlying type theory.

It treats sets, groupoids, and higher groupoids on an equal footing. One can easily remain entirely in the fragment of the theory which talks about sets, not worrying about groupoids or homotopy theory, but as soon as one starts to say something which naturally needs structures of higher homotopy level (such as talking about some collection of structured sets), the groupoidal and homotopical structure is already there.

New Axioms

As a foundation for mathematics whose basic objects are higher groupoids, homotopy type theory makes visible new foundational axioms. Most of these axioms are true as statements about classical ∞\infty-groupoids, but may be false in “nonclassical” models of homotopy type theory such as (∞,1)-toposes.

Additionally, since homotopy type theory as the internal language of (∞,1)(\infty, 1)-toposes is not classical, it is possible to add classically inconsistent axioms, just as, say, the Kock-Lawvere axiom may be added to ordinary topos theory.

denotes an object classifier in CC for a certain size of universe. Both Coq and Agda have systems to manage universe sizes and universe enlargement automatically; Agda’s is more advanced (universe polymorphism), whereas Coq’s is good enough for many purposes but tends to produce “universe inconsistencies” when working with univalence. As a stopgap measure until this is improved, some HoTT code must be compiled with a patch to Coq that turns off all universe consistency checks.

The first one is syntactic sugar for the second. The third is related to the second by eta expansion, which (assuming function extensionality) is an equivalence in Coq, but not the identity. In the next version of Coq, all three types above will be identical.

One might expect to also call the dependent sum type

exists x : X, P x

(see existential quantifier) but in the current Coq implementation that keyword is reserved for the corresponding operation on Coq’s built-in universe Prop, which is not used by homotopy type theory. In particular, it should not be confused with what HoTT calls propositions, which are the (-1)-truncated types. In fact, arguably in HoTT exists should refer not to the dependent sum itself, but to the (-1)-truncation thereof (its bracket type).

The fact that there is a weak equivalenceX→≃XIX \stackrel{\simeq}{\to} X^I given by the inclusion of identitymorphisms is reflected in the inductive type-definition of paths, which says that any proposition about terms in the path type is already determined by its value on all identity paths.

Then for x y : X two terms regarded as morphisms 1→X1\to X, the application of the identity type xx and yy (also called the identity type) denoted variously

paths X x y : Type
x ~~> y : Type
x == y : Type

(the latter two make use of Coq’s ability to define new notations), denotes the homotopy pullback of the form